PP Section 12.2

PP Section 12.2 - of polynomials. It can only be used when...

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Division of Polynomials

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Division Algorithm for Polynomials If f (x) and g (x) denote polynomial functions and if g (x) is not the zero polynomial, then there are unique polynomial functions q (x) and r (x) such that ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( x r x g x q x f or x g x r x q x g x f + = + = dividend quotient divisor remainder 5 1) - 3x(x 5 3x - 3 or 1 5 3 1 5 3 3 2 2 + = + - + = - + - x x x x x x Example:
Consider the function f(x) = 3x 3 + 9x 2 – 18 x – 24. Use the Division Algorithm to divide f(x) by g(x) = x + 3. 24 18 9 3 3 2 3 - - + + x x x x 3x 2 3x 3 + 9x 2 - 18x - 24 - 18 - 18x - 54 30 Subtract Subtract

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3 30 ) 18 3 ( 3 24 18 9 3 : Have 2 2 3 + + - = + - - + x x x x x x The quotient is 3x 2 – 18 and the remainder is 30 . 30 ) 3 )( 18 3 ( 24 18 9 3 : OR 2 2 3 + + - = - - + x x x x x
Use the Division Algorithm to divide f(x) by g(x) = x + 4. 24 18 9 3 4 2 3 - - + + x x x x 3x 2 3x 3 +12x 2 - 3x 2 - 18x - 24 - 3x - 3x 2 - 12x - 6x - 24 - 6 - 6x - 24 0 Subtract Subtract Subtract

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4 0 ) 6 3 3 ( 4 24 18 9 3 : Have 2 2 3 + + - - = + - - + x x x x x x x The quotient is 3x 2 – 3x - 6 and the remainder is 0. When the remainder is 0, we say that the divisor is a factor of the dividend.
Synthetic Division This is a simplified version of long division

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Unformatted text preview: of polynomials. It can only be used when the divisor has the form x a. The idea is to work only with the coefficients of the polynomials involved. From a previous example: 24 18 9 3 4 2 3--+ + x x x x 3x 2 3x 3 +12x 2- 3x 2- 18x - 24- 3x- 3x 2- 12x - 6x - 24- 6- 6x - 24 x + 4 = x (-4)- 4 3 9 - 18 - 24 3-12-3 12-6 24 Quotient = 3x 2 3x - 6 Remainder = 0 Example 2 by 5 6 divide o division t synthetic Use 3-+-x x x 2 1 0 -6 5 1 2 2 4-2-4 1 Quotient = x 2 +2x -2 Remainder = 1 Answer: x 3 6x + 5 = (x 2 +2x 2)(x - 2) + 1 Example ) 8 4 2 )( 2 ( 2 that show o division t synthetic Use 2 3 4 4 + + +-=-x x x x x 2 1 0 0 0 -16 1 2 2 4 4 8 8 16 Answer: x 4 2 4 = (x 3 +2x 2 + 4x + 8)(x - 2)...
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This note was uploaded on 01/22/2011 for the course MATH 2 taught by Professor Gardner during the Fall '08 term at Irvine Valley College.

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PP Section 12.2 - of polynomials. It can only be used when...

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